# FdeSolver.jl: Solving fractional differential equations

This is a **Julia** package for fractional differential equations and ODEs. It provides numerical solutions for nonlinear fractional ordinary differential equations (in the sense of Caputo).

Related work includes the independent FractionalDiffEq Julia Package that provides solutions of differential equations with different fractional operators, and earlier tools that are available in Matlab.

The implemented models are generic and broadly applicable to modeling multivariate signals from a single source or collected across multiple sources. The dynamical models implemented in this package were initially developed for modeling dynamics of interacting microbial communities (Khalighi et al. 2022) but the models are more broadly applicable and applicable to studying multi-omic and host-microbiome interactions.

## Method

We implement the predictor-corrector algorithms with a sufficient convergence and accuracy, including fast Fourier transform technique that gives us high computation speed. Interested readers can also find the stability of the methods and see how to implement the methods for solving multi-term fractional differential equations.

Let us suppose the following initial value problem with the Caputo fractional derivative for $\beta>0$:

\[{}_{C}\!D_{t_0}^{\beta}y(t)=f(t,y(t))\]

with the initial condition:

\[y(t_0)=y_0,y^{(1)}(t_0)=y^{(1)}_0,...,y^{(m-1)}(t_0)=y^{(m-1)}_0\]

where m is the smallest integer or equal to the order of derivative.

We solve the problem by using predictor corrector and Newton Raphson method.

## Installation

If Julia is installed correctly, you can import FdeSolver.jl as:

`import Pkg; Pkg.add("FdeSolver")`

A few methods on its usage are explained in Examples.

## Acknowledgments

We are grateful to all contributors. New issues and pull requests are welcome.

This research has received funding from

- the Horizon 2020 Programme of the European Union within the FindingPheno project under grant agreement No 952914.
- Research Council of Finland (grant 330887)

Package documentation is compiled according to the guidelines provided in PkgTutorial.jl.

## Publications

**Kindly cite this work** as follows:

- Fdesolver: A Julia package for solving fractional differential equations M Khalighi, G Benedetti, L Lahti arXiv preprint, 2022. arXiv:2212.12550

The package development is further linked with the following publications/preprints:

Quantifying the impact of ecological memory on the dynamics of interacting communities. M Khalighi, G Sommeria-Klein, D Gonze, K Faust, L Lahti. PLoS Computational Biology 18(6), 2022 doi:10.1371/journal.pcbi.1009396

Three-species Lotka-Volterra model with respect to Caputo and Caputo-Fabrizio fractional operators. M Khalighi, L Eftekhari, S Hosseinpour, L Lahti. Symmetry 13 (3):368, 2021. doi:10.3390/sym13030368

Ebola epidemic model with dynamic population and memory, F Ndaïrou, M Khalighi, and L Lahti, Chaos, Solitons \& Fractals, 170: 113361, 2023. doi:10.1016/j.chaos.2023.113361